1. Field of the Invention
The present invention relates to a thermal analysis instrument, and more particularly, to a differential scanning calorimeter.
2. Background of the Invention
Heat Flux Differential Scanning Calorimeters (DSCs) have a sensor which measures the temperature difference between a sample and a reference position. A sample to be analyzed is loaded into a pan and placed on the sample position of the sensor and an inert reference material is loaded into a pan and placed on the reference position of the sensor (an empty pan is often used as the reference). The sensor is installed in an oven whose temperature is varied dynamically according to a desired temperature program. The temperature program for conventional DSCs typically includes combinations of linear temperature ramps and constant temperature segments. Modulated DSC uses a temperature program in which periodic temperature modulations are superimposed on linear ramps. Modulated DSCs are described in U.S. Pat. No. 5,224,775, which is incorporated by reference herein. During the dynamic portion of the DSC experiment, a differential temperature is created between the sample and reference positions on the sensor. The temperature difference is the result of the difference between the heat flow to the sample and the heat flow to the reference. Because the temperature difference is proportional to the difference in heat flow to the sample as compared to the reference, that differential temperature may be used to measure the heat flow to the sample.
FIG. 1 shows a thermal network model that may be used to represent heat flux in certain DSC sensors. To is the temperature at the base of the sensor near its connection to the oven, Ts is the temperature of the sample position of the sensor and Tr is the temperature of the reference position of the sensor. Rs and Rr represent the thermal resistance between the base of the sensor and the sample and reference positions, respectively. Cs and Cr represent the thermal capacitance of the sample and reference portions of the sensor. Thermal capacitance is the product of mass and specific heat and is a measure of the heat storage capacity of a body, i.e., it is the heat capacity of the body. The heat flow to the sample and the reference are qs and qr, respectively. It should be understood that qs and qr include heat flow to sample and reference pans. During the execution of a thermal program the base temperature of the sensor To follows the thermal program. The temperatures at the sample and reference positions, Ts and Tr, lag the base temperature To due to heat flowing to the sample and to the reference and heat which is stored within the sensor in sensor sample thermal capacitance Cs and sensor reference thermal capacitance Cr.
Performing a heat flow balance on the sample side of the sensor yields a heat flow       q    s    =                              T          o                -                  T          s                            R        s              -                  C        s            ·                        ⅆ                      T            s                                    ⅆ          τ                    
trough the sensor sample thermal resistance Rs minus the heat stored in Cs. Similarly, a heat balance on the reference side of the sensor gives       q    r    =                              T          o                -                  T          r                            R        r              -                  C        r            ·                        ⅆ                      T            r                                    ⅆ          τ                    
through sensor reference thermal resistance Rr minus the heat stored in Cr. In the equations herein, xcfx84 represents time.
The desired quantity (the differential heat flow to the sample with respect to the reference) is the difference between the sample and reference heat flows:
q=qsxe2x88x92qr
Substituting for qs and qr yields:   q  =                              T          o                -                  T          s                            R        s              -                  C        s            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                            T          o                -                  T          r                            R        r              +                  C        r            ·                        ⅆ                      T            r                                    ⅆ          τ                    
Substituting the following expressions for two temperature differences in a differential scanning calorimeter,
xcex94T=Tsxe2x88x92Tr
xcex94To=Toxe2x88x92Ts
where xcex94T is the temperature difference between the sample and the reference and xcex94To is the temperature difference between the sample and a position at the base of the sensor, results in the DSC heat flow equation:   q  =            Δ      ⁢              xe2x80x83            ⁢                        T          o                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C                          s              ⁢                              xe2x80x83                                                    )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -          Cr      ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The DSC heat flow equation has 4 terms. The first term accounts for the effect of the difference between the sensor sample thermal resistance and the sensor reference thermal resistance. The second term is the conventional DSC heat flow. The third term accounts for the effect of the difference between the sensor sample thermal capacitance and the sensor reference thermal capacitance. The fourth term accounts for the effect of the difference between the heating rates of the sample and reference sides of the DSC. Conventionally, when this equation is applied to the DSC heat flow, the first and third terms are zero because Rs and Rr are assumed to be equal and Cs and Cr are also assumed to be equal.
In reality, because of imprecision in the manufacturing process, sensors are not perfectly balanced. This imbalance contributes to baseline heat flow deviations that may be significant. The first and third terms of the four-term heat flow equation account for the thermal resistance and thermal capacitance imbalances, respectively. The fourth term is generally very nearly equal to zero, except when a transition is occurring in the sample (for instance, during a melt), or during a Modulated DSC experiment. Usually, the heat flow signal is integrated over the area of the transition to obtain the total energy of the transition. Because the fourth term does not contribute to the area of the integration, it has been ignored in the prior art. However, it may contribute significantly to the shape of the heat flow curve during a transition. Therefore, including the fourth term improves the dynamic response of the heat flow curve. Also, as noted by Hohne, et. al. in xe2x80x9cDifferential Scanning Calorimetry: An Introduction for Practitioners,xe2x80x9d (Springer-Verlag, 1996), the fourth term cannot be ignored and must be taken into account when partial integration of the transition peak is performed, e.g., in kinetic investigations for purity determinations. When the fourth term is included, the onset of a transition is sharper and the return to baseline heat flow when the transition is over is more rapid.
Because the resolution of a DSC is its ability to separate transitions that occur in a sample within a small temperature interval, and this is determined essentially by how quickly the heat flow signal returns to baseline after a transition is complete, including the fourth term of the DSC heat flow equation improves the resolution of the DSC sensor by increasing the return to baseline of the heat flow signal after a transition is complete.
The four-term heat flow equation has long been known in the art of differential scanning calorimetry. It can only be applied to heat flux DSC sensors that satisfy certain criteria. The structure of the sensor must be such that the thermal network model correctly represents the dynamic thermal behavior of sensor. Ideally, the sample and reference portions of the sensor should be absolutely independent, i.e., a transition that occurs on the sample side would not have any effect on the reference temperature. Typically, heat flux DSC sensors of the disk type as disclosed in U.S. Pat. No. 4,095,453 to Woo, U.S. Pat. No. 4,350,446 to Johnson, U.S. Pat. No. 5,033,866 to Kehl, et. al. and U.S. Pat. No. 5,288,147 to Schaefer, et. al. cannot use the four-term heat flow equation, because the sample and reference temperature are not independent, and because the four-term heat flow equation does not accurately represent the dynamic thermal behavior of those sensors.
A quantitative measure of the independence of prior art heat flux sensors can be obtained by a simple experiment: for example, if a sample of indium is placed on the reference position of a prior art sensor such as, for example, the type disclosed in U.S. Pat. No. 4,095,453 to Woo, and the sample is heated through the melt, in one exemplary experiment the deviation of the temperature of the sample position observed was observed to be 13.4% of the deviation that would have been obtained if the indium sample had been placed on the sample position. In an ideal instrument, that deviation would have been zero.
U.S. Pat. No. 5,599,104 to Nakamura, et. al. discloses a heat flux DSC sensor that uses two temperature difference measurements. However, these measurements are applied in a different manner using a different heat flow equation and the configuration of the differential temperature measurements is not suitable for use with the four-term heat flow equation. Specifically, Nakamura cannot use the four-term equation because the two temperature differences measured in Nakamura are not suitable for use in the four-term DSC the equation.
The present invention is a differential scanning calorimeter sensor that measures the differential heat flow to the sample based upon a single absolute temperature measurement and two differential temperature measurements. Differential scanning calorimeters of the present invention have substantially improved resolution over conventional instruments, with an empty-cell heat flow that is much closer to zero than that obtained in conventional instruments.
Temperature Measurements
In the present invention, the differential heat flow to the sample with respect to the reference is calculated from measurements of the absolute temperature of the base of the sensor, the differential temperature between the sample position and the base of the sensor, and the differential temperature between the sample and reference positions. The differential temperatures are measured using a sample area temperature detector and a reference area temperature detector.
The base temperature detector (which measures the temperature of the base of the sensor near its connection to the oven) is used to control the oven temperature. The sample temperature is measured by measuring the difference between the sample temperature and the base temperature, and subtracting that difference from the base temperature to obtain the sample temperature, i.e., the sample temperature is obtained from Ts=Toxe2x88x92xcex94Ts. By making a single absolute temperature measurement, To and two differential temperature measurements, any relative errors in absolute temperature measurements due to differences in temperature sensors are eliminated. Also, this structure minimizes the drift of sample temperature during isothermal segments.
The sensor constructed according to the present invention exhibits improved independence between the sample and reference positions. For example, whereas in the exemplary experiment described in the background section, in a prior art heat flux instrument a deviation in the temperature of the sample position of 13.4% was observed when a sample of indium is placed on the reference position and heated through the melt, with the present invention, in an exemplary experiment with the present invention that deviation is only about 1.4%, i.e., the present invention exhibits an improvement by about an order of magnitude over the prior art instruments. Thus sensors constructed according to the present invention are xe2x80x9ceffectively independent,xe2x80x9d because they exhibit a temperature deviation at the sample position when an indium sample is placed on the reference position of less than about 1.5% of the temperature deviation at the sample position when an indium sample is placed on the sample position.
Calibration
In a first preferred embodiment, the differential scanning calorimeter of the present invention is calibrated by running two separate experiments. These experiments determine the four sensor thermal parameters, Cs (the sensor sample thermal capacitance), Cr (the sensor reference thermal capacitance), Rs (the sensor sample thermal resistance) and Rr (the sensor reference thermal resistance) experimentally, and thus calibrate the heat flow sensor.
The first experiment is performed with an empty DSC cell. The DSC cell is first held at an isothermal temperature that is below the temperature range of the calibration, for a time segment sufficient to ensure complete equilibration of the sensor. The DSC cell is then heated at a constant heating rate to a temperature above the temperature range of the calibration, and then held at that temperature for another isothermal segment, for a time segment sufficient to ensure equilibration of the sensor at that temperature. This first experiment is used to calculate the sample and reference time constants as a function of temperature over the calibrated temperature range.
The heat flow to the sample and the heat flow to the reference should be zero (since the DSC cell is empty). Accordingly, if qs and qr are set equal to zero in the heat balance equations for the sample and reference sides of the sensor, the time constants for the sample and reference are given by:             τ      s        =                            C          s                ⁢                  R          s                    =                                    Δ            ⁢                          xe2x80x83                        ⁢                          T              o                                            (                                          ⅆ                                  T                  s                                                            ⅆ                τ                                      )                          ⁢                  xe2x80x83                ⁢        and                        τ      r        =                            C          r                ⁢                  R          r                    =                                    Δ            ⁢                          xe2x80x83                        ⁢                          T              o                                +                      Δ            ⁢                          xe2x80x83                        ⁢            T                                                              ⅆ                              T                s                                                    ⅆ              τ                                -                                                    ⅆ                Δ                            ⁢                              xe2x80x83                            ⁢              T                                      ⅆ              τ                                          
respectively. These results are stored as a function of temperature.
The second experiment uses a pair of calibration samples. The calibration samples may have the same mass, or may have different masses. Preferably, the calibration samples are sapphire samples, preferably weighing 25 mg or more. Other reference materials with well-known specific heats and no transitions in the temperature range of the calibration may be used instead of sapphire (in which case Csapph would be replaced in the following equations by Cmat where Cmat is the specific heat of the other reference material).
The sample and reference heat flows from the heat balance equations are set as follows:             q      s        =                  m        s            ·              C        sapph            ·                        ⅆ                      T            ss                                    ⅆ          τ                                q      r        =                  m        r            ·              C        sapph            ·                        ⅆ                      T            rs                                    ⅆ          τ                    
where ms, mr are the masses of the sample and reference sapphires, respectively, Csapph is the specific heat of sapphire and Tss and Trs are the temperatures of the sample and reference sapphire.
Assume:             ⅆ              T        ss                    ⅆ      τ        =                              ⅆ                      T            s                                    ⅆ          τ                    ⁢              xe2x80x83            ⁢      and      ⁢              xe2x80x83            ⁢                        ⅆ                      T            rs                                    ⅆ          τ                      =                  ⅆ                  T          r                            ⅆ        τ            
Substituting for qs and Ts in the sample heat balance equation and solving for the sensor sample thermal capacitance Cs:       C    s    =                    m        s            ·      Csapph                                Δ          ⁢                      xe2x80x83                    ⁢                      T            o                                                (                                          ⅆ                                  T                  s                                                            ⅆ                τ                                      )                    ·                      τ            s                              -      1      
Substituting for qs and xcfx84r in the reference heat balance equation and solving for the sensor reference thermal capacitance Cr:       C    r    =                    m        r            ·              C        sapph                                                  Δ            ⁢                          xe2x80x83                        ⁢                          T              o                                +                      Δ            ⁢                          xe2x80x83                        ⁢            T                                                (                                                            ⅆ                                      T                    s                                                                    ⅆ                  τ                                            -                                                                    ⅆ                    Δ                                    ⁢                                      xe2x80x83                                    ⁢                  T                                                  ⅆ                  τ                                                      )                    ·                      τ            r                              -      1      
The results from the second experiment using sapphire (or another well-known calibration material; if the calibration material is not sapphire, then replace Csapph in the equations herein with Cmat, the specific heat of the calibration material) using the time constants for DSC cell obtained in the first experiment are then used to calculate the sample and reference sensor heat capacities as a function of temperature. Finally, the sensor sample and reference thermal resistances are calculated from the time constants and the sensor thermal capacitances:       R    s    =                              τ          s                          C          s                    ⁢              xe2x80x83            ⁢      and      ⁢              xe2x80x83            ⁢              R        r              =                  τ        r                    C        r            
A second preferred embodiment is similar to the first embodiment, but uses sapphire (or another material with a well-known heat capacity and no transitions in the temperature range of interest) calibration samples in both the first and the second calibration experiments. The calibration equations and their derivation for this embodiment are described below.
Accordingly, the present invention is a differential scanning calorimeter that provides improved resolution by accounting for all four terms of the heat flow equation. The present invention also provides a differential scanning calorimeter with an empty-cell heat flow much closer to zero than conventional instruments. Another advantage of the present invention is that it provides a more complete and correct measure of the heat flow to the sample.